Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C53 ⋊ #11912 |
284 conjugacy classes of subgroups; only the 83 normal subgroups are shown.
This group has 284 conjugacy classes of subgroups — too many to draw. Only its 83 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 1696 |
A minimal faithful permutation representation — 3 generators on 85 points:
(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(54,56)(55,61,64,77)(57,70)(58,68)(59,66)(60,65,62,78)(63,81,80,76)(67,71)(69,84)(72,75)(73,83)(74,85,79,82);(55,60)(57,69)(58,73)(61,65)(62,64)(63,80)(67,75)(68,83)(70,84)(71,72)(76,81)(77,78);(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25)(26,27)(28,29)(30,31)(32,33)(34,35)(36,37)(38,39)(40,41)(42,43)(44,45)(46,47)(48,49)(50,51)(52,53)(54,55,57,63,58,60,72,74,59,64,71,80,73,62,69,79)(56,65,67,82,68,77,84,81,66,78,70,85,83,61,75,76)Polycyclic presentation:
2^8,53|a^2=e,c^2=g,d^2=g,e^2=f,f^2=h,b^a=bd,c^a=cf,d^a=df,g^a=gh,i^a=i^52,d^b=dg,e^b=efgh,f^b=fh,d^c=dh,e^c=eh,e^d=eh866 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 80750);